Hi, I'm new to the group, but I have read the entire archive. I noticed
rather little work done on Square 1. It seems to me that this puzzle
deserves a closer look for finding God's algorithm. Mike Reid's
calculations notwithstanding (archive 17), I have found that the problem
can be reduced by at least a factor of 400 if we just get rid of
combinations that result from trivial face turns, and if we note that
the Start position has a degeneracy of 16. (One center slice is assumed
fixed - another factor of 2 is tempting but not possible)

Mike's calcualtion for the number of states would reduce to:
 (2*(1/6)*(9/2)+2*(28/3)*3+(35/4)*(35/4))*2*8!*8!=435891456000
combinations.  Divide by the start degeneracy, multiply by 2 storage
bits per state, and you get a storage requirement of 6.81GB.  This seems
very close to being doable.  Maybe in another 10 years, I can do this
project on my PC, if no one has done it yet. 

On another note, when I signed up, I mentioned to Alan that I must be
crazy enough to join this group since I have a five foot mockup of a
rubik's type puzzle as my coffee table. He thought its description
might be of general interest. Skip the rest of this paragraph if you
couldn't care less about its origins. I built it for Caltech's ditch day event
Maybe you have heard of it. That's where all the seniors leave for the
day with their room locked only with a puzzle of some sort, and the
object is for the undergraduates to get into the room by solving it
(with a couple of clues, of course). Anyway, being as it was that I
had a mechanical engineer roommate... The rest is history, and I now
have a five foot diameter puzzle coffee table.

OK, a description. The puzzle is a three centered version of the
Puzzler, widely available in the last few years in puzzle/game specialty
stores. The differences being that it is colored so that the maximum
number of combinatins are possible (including the supergroup of distinguishing
face centers). For those of you who have not seen the Puzzler, and thus
have no frame of reference, consider one vertex of a cube and it's
surrounding faces. 7 vertices, 9 edges. Faces can undergo 4 quarter
turns. Extrapolate to the Megaminx and you again get one central vertex
for a total of 10 vertices and 12 edges. Faces can undergo 1/5 turns.
Extrapolate again to six sided faces, and you get a flat puzzle with one
central vertex for a total of 13 vertices, and 15 edges. Faces can
undergo 1/6 turns.

So it is basically the <URF> group for a' cube' with
hexagonal faces. The extra face over the Puzzler also serves to remove
the significant parity constraints on the edge pieces. (compare <UR>
group to <URF> group of regular cube).

You, too, can make a smaller version of the Ditch Day puzzle at home.
The advantage of the flat puzzle is that it is easily constructed. I
built the 6 inch diameter prototype with poster board, lamination, magic
markers, and an easily machined smooth pressboard frame. You only need
to drill three 3 inch holes. The rest is trimming. Oh yeah, you will
need a plexiglass faceplate to keep the pieces in too. Cutting out and
gluing together the poster board to make sufficiently thick pieces was
the hardest part.

Number of combinations = (13!*15!*3^13*2^15*6^3)/24 = 3.83E33
Difficulty is comparable with Megaminx.

Happy cubing. This is already too long.

mikem.
Mike Masonjones.